Bilge, Ayşe HümeyraKumbasar, YalcinBilge, Ayşe Hümeyra2019-06-272019-06-27201200024-37950024-3795https://hdl.handle.net/20.500.12469/955https://doi.org/10.1016/j.laa.2011.06.034It is well known that a commuting family of diagonalizable linear operators on a finite dimensional vector space is simultaneously diagonalizable. In this paper we consider a family A = {A(a)} A(a) : V -> V a = 1... N of anti-commuting (complex) linear operators on a finite dimensional vector space. We prove that if the family is diagonalizable over the complex numbers then V has an A-invariant direct sum decomposition into subspaces V(alpha) such that the restriction of the family A to V(alpha) is a representation of a Clifford algebra. Thus unlike the families of commuting diagonalizable operators diagonalizable anti-commuting families cannot be simultaneously digonalized but on each subspace they can be put simultaneously to (non-unique) canonical forms. The construction of canonical forms for complex representations is straightforward while for the real representations it follows from the results of [A.H. Bilge S. Kocak S. Uguz Canonical bases for real representations of Clifford algebras Linear Algebra Appl. 419 (2006) 417-439]. (C) 2011 Elsevier Inc. All rights reserved.eninfo:eu-repo/semantics/openAccessAnti-commuting linear operatorsRepresentations of Clifford algebrasArticle79851436WOS:00029743120000810.1016/j.laa.2011.06.0342-s2.0-80055063962Q2Q1